I was looking at a proof for Newton Sums (Proof), and it concludes the following:
So if you were to have a polynomial with four terms and four non-zero roots the following should be true: $a_3P_1+a_2P_0+a_1P_{-1}+a_0P_{-2} = 0$, but the site claims that $a_3P_1+a_2 = 0$. Are root sums with powers less than or equal to zero not supposed to be considered in the equation? Why is that the case?
A simplified polynomial with four terms has degree $3$. Hence there can only be three non-zero roots. So you would have: $$a_3P_1+3a_2+a_1P_{-1}+a_0P_{-2}=0.$$ Since $a_3P_1+a_2=0$ (which is a rearrangement of Vieta's formula), we must have: $$2a_2+a_1P_{-1}+a_0P_{-2}=0.$$ Nothing in the proof indicates that root sums with powers less than or equal to zero cannot be considered.